Theorem mapdpglem3

Description: Lemma for mapdpg . Baer p. 45, line 3: "infer ... the existence of a number g in G and of an element z in (Fy)* such that t = gx'-z." (We scope $d g w z ph locally to avoid clashes with later substitutions into ph .) (Contributed by NM, 18-Mar-2015)

	Ref	Expression
Hypotheses	mapdpglem.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
	mapdpglem.m	⊢ 𝑀 = ( ( mapd ‘ 𝐾 ) ‘ 𝑊 )
	mapdpglem.u	⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
	mapdpglem.v	⊢ 𝑉 = ( Base ‘ 𝑈 )
	mapdpglem.s	⊢ − = ( -g ‘ 𝑈 )
	mapdpglem.n	⊢ 𝑁 = ( LSpan ‘ 𝑈 )
	mapdpglem.c	⊢ 𝐶 = ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 )
	mapdpglem.k	⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
	mapdpglem.x	⊢ ( 𝜑 → 𝑋 ∈ 𝑉 )
	mapdpglem.y	⊢ ( 𝜑 → 𝑌 ∈ 𝑉 )
	mapdpglem1.p	⊢ ⊕ = ( LSSum ‘ 𝐶 )
	mapdpglem2.j	⊢ 𝐽 = ( LSpan ‘ 𝐶 )
	mapdpglem3.f	⊢ 𝐹 = ( Base ‘ 𝐶 )
	mapdpglem3.te	⊢ ( 𝜑 → 𝑡 ∈ ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑋 } ) ) ⊕ ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) ) )
	mapdpglem3.a	⊢ 𝐴 = ( Scalar ‘ 𝑈 )
	mapdpglem3.b	⊢ 𝐵 = ( Base ‘ 𝐴 )
	mapdpglem3.t	⊢ · = ( ·𝑠 ‘ 𝐶 )
	mapdpglem3.r	⊢ 𝑅 = ( -g ‘ 𝐶 )
	mapdpglem3.g	⊢ ( 𝜑 → 𝐺 ∈ 𝐹 )
	mapdpglem3.e	⊢ ( 𝜑 → ( 𝑀 ‘ ( 𝑁 ‘ { 𝑋 } ) ) = ( 𝐽 ‘ { 𝐺 } ) )
Assertion	mapdpglem3	⊢ ( 𝜑 → ∃ 𝑔 ∈ 𝐵 ∃ 𝑧 ∈ ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) 𝑡 = ( ( 𝑔 · 𝐺 ) 𝑅 𝑧 ) )

Proof

Step	Hyp	Ref	Expression
1		mapdpglem.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
2		mapdpglem.m	⊢ 𝑀 = ( ( mapd ‘ 𝐾 ) ‘ 𝑊 )
3		mapdpglem.u	⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
4		mapdpglem.v	⊢ 𝑉 = ( Base ‘ 𝑈 )
5		mapdpglem.s	⊢ − = ( -g ‘ 𝑈 )
6		mapdpglem.n	⊢ 𝑁 = ( LSpan ‘ 𝑈 )
7		mapdpglem.c	⊢ 𝐶 = ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 )
8		mapdpglem.k	⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
9		mapdpglem.x	⊢ ( 𝜑 → 𝑋 ∈ 𝑉 )
10		mapdpglem.y	⊢ ( 𝜑 → 𝑌 ∈ 𝑉 )
11		mapdpglem1.p	⊢ ⊕ = ( LSSum ‘ 𝐶 )
12		mapdpglem2.j	⊢ 𝐽 = ( LSpan ‘ 𝐶 )
13		mapdpglem3.f	⊢ 𝐹 = ( Base ‘ 𝐶 )
14		mapdpglem3.te	⊢ ( 𝜑 → 𝑡 ∈ ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑋 } ) ) ⊕ ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) ) )
15		mapdpglem3.a	⊢ 𝐴 = ( Scalar ‘ 𝑈 )
16		mapdpglem3.b	⊢ 𝐵 = ( Base ‘ 𝐴 )
17		mapdpglem3.t	⊢ · = ( ·𝑠 ‘ 𝐶 )
18		mapdpglem3.r	⊢ 𝑅 = ( -g ‘ 𝐶 )
19		mapdpglem3.g	⊢ ( 𝜑 → 𝐺 ∈ 𝐹 )
20		mapdpglem3.e	⊢ ( 𝜑 → ( 𝑀 ‘ ( 𝑁 ‘ { 𝑋 } ) ) = ( 𝐽 ‘ { 𝐺 } ) )
21	20	oveq1d	⊢ ( 𝜑 → ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑋 } ) ) ⊕ ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) ) = ( ( 𝐽 ‘ { 𝐺 } ) ⊕ ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) ) )
22	14 21	eleqtrd	⊢ ( 𝜑 → 𝑡 ∈ ( ( 𝐽 ‘ { 𝐺 } ) ⊕ ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) ) )
23		r19.41v	⊢ ( ∃ 𝑔 ∈ 𝐵 ( 𝑤 = ( 𝑔 · 𝐺 ) ∧ ∃ 𝑧 ∈ ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) 𝑡 = ( 𝑤 𝑅 𝑧 ) ) ↔ ( ∃ 𝑔 ∈ 𝐵 𝑤 = ( 𝑔 · 𝐺 ) ∧ ∃ 𝑧 ∈ ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) 𝑡 = ( 𝑤 𝑅 𝑧 ) ) )
24	1 7 8	lcdlmod	⊢ ( 𝜑 → 𝐶 ∈ LMod )
25		eqid	⊢ ( Scalar ‘ 𝐶 ) = ( Scalar ‘ 𝐶 )
26		eqid	⊢ ( Base ‘ ( Scalar ‘ 𝐶 ) ) = ( Base ‘ ( Scalar ‘ 𝐶 ) )
27	25 26 13 17 12	lspsnel	⊢ ( ( 𝐶 ∈ LMod ∧ 𝐺 ∈ 𝐹 ) → ( 𝑤 ∈ ( 𝐽 ‘ { 𝐺 } ) ↔ ∃ 𝑔 ∈ ( Base ‘ ( Scalar ‘ 𝐶 ) ) 𝑤 = ( 𝑔 · 𝐺 ) ) )
28	24 19 27	syl2anc	⊢ ( 𝜑 → ( 𝑤 ∈ ( 𝐽 ‘ { 𝐺 } ) ↔ ∃ 𝑔 ∈ ( Base ‘ ( Scalar ‘ 𝐶 ) ) 𝑤 = ( 𝑔 · 𝐺 ) ) )
29	1 3 15 16 7 25 26 8	lcdsbase	⊢ ( 𝜑 → ( Base ‘ ( Scalar ‘ 𝐶 ) ) = 𝐵 )
30	29	rexeqdv	⊢ ( 𝜑 → ( ∃ 𝑔 ∈ ( Base ‘ ( Scalar ‘ 𝐶 ) ) 𝑤 = ( 𝑔 · 𝐺 ) ↔ ∃ 𝑔 ∈ 𝐵 𝑤 = ( 𝑔 · 𝐺 ) ) )
31	28 30	bitrd	⊢ ( 𝜑 → ( 𝑤 ∈ ( 𝐽 ‘ { 𝐺 } ) ↔ ∃ 𝑔 ∈ 𝐵 𝑤 = ( 𝑔 · 𝐺 ) ) )
32	31	anbi1d	⊢ ( 𝜑 → ( ( 𝑤 ∈ ( 𝐽 ‘ { 𝐺 } ) ∧ ∃ 𝑧 ∈ ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) 𝑡 = ( 𝑤 𝑅 𝑧 ) ) ↔ ( ∃ 𝑔 ∈ 𝐵 𝑤 = ( 𝑔 · 𝐺 ) ∧ ∃ 𝑧 ∈ ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) 𝑡 = ( 𝑤 𝑅 𝑧 ) ) ) )
33	23 32	bitr4id	⊢ ( 𝜑 → ( ∃ 𝑔 ∈ 𝐵 ( 𝑤 = ( 𝑔 · 𝐺 ) ∧ ∃ 𝑧 ∈ ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) 𝑡 = ( 𝑤 𝑅 𝑧 ) ) ↔ ( 𝑤 ∈ ( 𝐽 ‘ { 𝐺 } ) ∧ ∃ 𝑧 ∈ ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) 𝑡 = ( 𝑤 𝑅 𝑧 ) ) ) )
34	33	exbidv	⊢ ( 𝜑 → ( ∃ 𝑤 ∃ 𝑔 ∈ 𝐵 ( 𝑤 = ( 𝑔 · 𝐺 ) ∧ ∃ 𝑧 ∈ ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) 𝑡 = ( 𝑤 𝑅 𝑧 ) ) ↔ ∃ 𝑤 ( 𝑤 ∈ ( 𝐽 ‘ { 𝐺 } ) ∧ ∃ 𝑧 ∈ ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) 𝑡 = ( 𝑤 𝑅 𝑧 ) ) ) )
35		df-rex	⊢ ( ∃ 𝑤 ∈ ( 𝐽 ‘ { 𝐺 } ) ∃ 𝑧 ∈ ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) 𝑡 = ( 𝑤 𝑅 𝑧 ) ↔ ∃ 𝑤 ( 𝑤 ∈ ( 𝐽 ‘ { 𝐺 } ) ∧ ∃ 𝑧 ∈ ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) 𝑡 = ( 𝑤 𝑅 𝑧 ) ) )
36	34 35	bitr4di	⊢ ( 𝜑 → ( ∃ 𝑤 ∃ 𝑔 ∈ 𝐵 ( 𝑤 = ( 𝑔 · 𝐺 ) ∧ ∃ 𝑧 ∈ ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) 𝑡 = ( 𝑤 𝑅 𝑧 ) ) ↔ ∃ 𝑤 ∈ ( 𝐽 ‘ { 𝐺 } ) ∃ 𝑧 ∈ ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) 𝑡 = ( 𝑤 𝑅 𝑧 ) ) )
37		eqid	⊢ ( LSubSp ‘ 𝐶 ) = ( LSubSp ‘ 𝐶 )
38	37	lsssssubg	⊢ ( 𝐶 ∈ LMod → ( LSubSp ‘ 𝐶 ) ⊆ ( SubGrp ‘ 𝐶 ) )
39	24 38	syl	⊢ ( 𝜑 → ( LSubSp ‘ 𝐶 ) ⊆ ( SubGrp ‘ 𝐶 ) )
40	13 37 12	lspsncl	⊢ ( ( 𝐶 ∈ LMod ∧ 𝐺 ∈ 𝐹 ) → ( 𝐽 ‘ { 𝐺 } ) ∈ ( LSubSp ‘ 𝐶 ) )
41	24 19 40	syl2anc	⊢ ( 𝜑 → ( 𝐽 ‘ { 𝐺 } ) ∈ ( LSubSp ‘ 𝐶 ) )
42	39 41	sseldd	⊢ ( 𝜑 → ( 𝐽 ‘ { 𝐺 } ) ∈ ( SubGrp ‘ 𝐶 ) )
43		eqid	⊢ ( LSubSp ‘ 𝑈 ) = ( LSubSp ‘ 𝑈 )
44	1 3 8	dvhlmod	⊢ ( 𝜑 → 𝑈 ∈ LMod )
45	4 43 6	lspsncl	⊢ ( ( 𝑈 ∈ LMod ∧ 𝑌 ∈ 𝑉 ) → ( 𝑁 ‘ { 𝑌 } ) ∈ ( LSubSp ‘ 𝑈 ) )
46	44 10 45	syl2anc	⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑌 } ) ∈ ( LSubSp ‘ 𝑈 ) )
47	1 2 3 43 7 37 8 46	mapdcl2	⊢ ( 𝜑 → ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) ∈ ( LSubSp ‘ 𝐶 ) )
48	39 47	sseldd	⊢ ( 𝜑 → ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) ∈ ( SubGrp ‘ 𝐶 ) )
49	18 11 42 48	lsmelvalm	⊢ ( 𝜑 → ( 𝑡 ∈ ( ( 𝐽 ‘ { 𝐺 } ) ⊕ ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) ) ↔ ∃ 𝑤 ∈ ( 𝐽 ‘ { 𝐺 } ) ∃ 𝑧 ∈ ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) 𝑡 = ( 𝑤 𝑅 𝑧 ) ) )
50	36 49	bitr4d	⊢ ( 𝜑 → ( ∃ 𝑤 ∃ 𝑔 ∈ 𝐵 ( 𝑤 = ( 𝑔 · 𝐺 ) ∧ ∃ 𝑧 ∈ ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) 𝑡 = ( 𝑤 𝑅 𝑧 ) ) ↔ 𝑡 ∈ ( ( 𝐽 ‘ { 𝐺 } ) ⊕ ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) ) ) )
51	22 50	mpbird	⊢ ( 𝜑 → ∃ 𝑤 ∃ 𝑔 ∈ 𝐵 ( 𝑤 = ( 𝑔 · 𝐺 ) ∧ ∃ 𝑧 ∈ ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) 𝑡 = ( 𝑤 𝑅 𝑧 ) ) )
52		ovex	⊢ ( 𝑔 · 𝐺 ) ∈ V
53		oveq1	⊢ ( 𝑤 = ( 𝑔 · 𝐺 ) → ( 𝑤 𝑅 𝑧 ) = ( ( 𝑔 · 𝐺 ) 𝑅 𝑧 ) )
54	53	eqeq2d	⊢ ( 𝑤 = ( 𝑔 · 𝐺 ) → ( 𝑡 = ( 𝑤 𝑅 𝑧 ) ↔ 𝑡 = ( ( 𝑔 · 𝐺 ) 𝑅 𝑧 ) ) )
55	54	rexbidv	⊢ ( 𝑤 = ( 𝑔 · 𝐺 ) → ( ∃ 𝑧 ∈ ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) 𝑡 = ( 𝑤 𝑅 𝑧 ) ↔ ∃ 𝑧 ∈ ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) 𝑡 = ( ( 𝑔 · 𝐺 ) 𝑅 𝑧 ) ) )
56	52 55	ceqsexv	⊢ ( ∃ 𝑤 ( 𝑤 = ( 𝑔 · 𝐺 ) ∧ ∃ 𝑧 ∈ ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) 𝑡 = ( 𝑤 𝑅 𝑧 ) ) ↔ ∃ 𝑧 ∈ ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) 𝑡 = ( ( 𝑔 · 𝐺 ) 𝑅 𝑧 ) )
57	56	rexbii	⊢ ( ∃ 𝑔 ∈ 𝐵 ∃ 𝑤 ( 𝑤 = ( 𝑔 · 𝐺 ) ∧ ∃ 𝑧 ∈ ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) 𝑡 = ( 𝑤 𝑅 𝑧 ) ) ↔ ∃ 𝑔 ∈ 𝐵 ∃ 𝑧 ∈ ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) 𝑡 = ( ( 𝑔 · 𝐺 ) 𝑅 𝑧 ) )
58		rexcom4	⊢ ( ∃ 𝑔 ∈ 𝐵 ∃ 𝑤 ( 𝑤 = ( 𝑔 · 𝐺 ) ∧ ∃ 𝑧 ∈ ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) 𝑡 = ( 𝑤 𝑅 𝑧 ) ) ↔ ∃ 𝑤 ∃ 𝑔 ∈ 𝐵 ( 𝑤 = ( 𝑔 · 𝐺 ) ∧ ∃ 𝑧 ∈ ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) 𝑡 = ( 𝑤 𝑅 𝑧 ) ) )
59	57 58	bitr3i	⊢ ( ∃ 𝑔 ∈ 𝐵 ∃ 𝑧 ∈ ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) 𝑡 = ( ( 𝑔 · 𝐺 ) 𝑅 𝑧 ) ↔ ∃ 𝑤 ∃ 𝑔 ∈ 𝐵 ( 𝑤 = ( 𝑔 · 𝐺 ) ∧ ∃ 𝑧 ∈ ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) 𝑡 = ( 𝑤 𝑅 𝑧 ) ) )
60	51 59	sylibr	⊢ ( 𝜑 → ∃ 𝑔 ∈ 𝐵 ∃ 𝑧 ∈ ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) 𝑡 = ( ( 𝑔 · 𝐺 ) 𝑅 𝑧 ) )